Buckling, Post-buckling, Collapse And Design Of Two-span Cold-formed .
SDSS’Rio 2010 STABILITY AND DUCTILITY OF STEEL STRUCTURESE. Batista, P. Vellasco, L. de Lima (Eds.)Rio de Janeiro, Brazil, September 8 - 10, 2010BUCKLING, POST-BUCKLING, COLLAPSE AND DESIGNOF TWO-SPAN COLD-FORMED STEEL BEAMSCilmar Basaglia and Dinar CamotimDepartment of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Portugale-mails: cbasaglia@civil.ist.utl.pt, dcamotim@civil.ist.utl.ptKeywords: Cold-formed steel, Two-span continuous beam, Buckling, Post-buckling, Structural designAbstract. This paper reports the available results of an ongoing numerical investigation on the buckling, postbuckling, collapse and design of two-span cold-formed steel lipped channel beams subjected to uniformlydistributed loads. The results presented and discussed are obtained through analyses based on Generalised BeamTheory (elastic buckling analyses) and shell finite element models (elastic and elastic-plastic post-buckling analysesup to collapse). Moreover, the ultimate loads obtained are used to establish preliminary guidelines concerning thedesign of continuous (multi-span) cold-formed steel beams failing in modes that combine local, distortional andglobal features. An approach based on the existing Direct Strength Method (DSM) expressions is followed and thecomparison between the numerical and predicted ultimate loads makes it possible to draw some conclusionsconcerning the issues that must be addressed by a DSM design procedure for cold-formed continuous beams.1 INTRODUCTIONIn order to adequately design and assess the structural efficiency of cold-formed steel (thin-walled) members onemust acquire in-depth knowledge on their non-linear response, a complex task that requires evaluating bucklingstresses and determining post-buckling equilibrium paths up to collapse (accounting for initial imperfections). Indeed,a fair amount of research work has been recently devoted to the development of efficient design rules forisolated thin-walled members. The most successful end product of this research activity was the increasinglypopular “Direct Strength Method” (DSM) [1], already included in the current Australian/New Zealander(AS/NZS4600: 2005) and North American (NAS: AISI-S100-07) specifications for cold-formed steel structures.In practice, many thin-walled structural members exhibit multiple spans (e.g., secondary elements likepurlins or side rails) and are often subjected to non-uniform bending moment diagrams that combine positive(sagging) and negative (hogging) regions, a feature making their buckling behaviour rather complex, as it often (i)combines local, distortional and global features and (ii) involves a fair amount of localisation (e.g., the occurrence oflocal and/or distortional buckling in the vicinity of the intermediate supports, where there are significant momentgradients and very little restraint can be offered to the slender bottom/compressed flanges). Even so, it seems fairto say that it is still very scarce the amount of research on the buckling and post-buckling behaviours of thin-walledsteel beams subjected to non-uniform bending moment diagrams, namely continuous beams. In this context, it isworth mentioning the recent works of (i) Camotim et al. [2], who used Generalised Beam Theory (GBT) to analysethe buckling behaviour of steel beams with distinct loadings and support conditions (including intermediate supports),and (ii) Yu and Schafer [3], who investigated the influence of a linear bending moment gradient on the distortionalbuckling and post-buckling behaviours of single-span cold-formed steel beams, and used their finding toexamine and extend the DSM design procedure for such members.The aim of this work is to present and discuss the results of an ongoing numerical investigation on the buckling,post-buckling, collapse and DSM design of two-span lipped channel beams. The numerical results presented wereobtained through (i) GBT buckling analyses and (ii) elastic and elastic-plastic shell finite element (SFE) postbuckling analyses. In particular, some interesting conclusions are drawn on the features that must be incorporated ina DSM design procedure for this type of cold-formed steel members.913
Cilmar Basaglia and Dinar Camotim2 NUMERICAL INVESTIGATION: SCOPE, MODELLING AND RESULTSThe buckling, post-buckling and ultimate strength results presented next concerning the non-linear behaviour ofsimply supported two-span symmetric lipped channel beams (i) with the cross-section dimensions given in figure1(a), (ii) having overall lengths of 4.0m (B4), 8.0m (B8) and 10.0m (B10), and (i) acted by a uniformly distributedload applied along the shear centre axis, causing only (pre-buckling) major-axis bending (see fig. 1(b)). The beamend sections are locally/globally pinned and can warp freely, and the intermediate support restrains all in-planecross-section displacements. The post-buckling analyses incorporate small-amplitude critical-mode initialimperfections and do not account for residual stresses.Concerning the GBT and SFE analyses, the following modelling issues deserve to be mentioned:(i) GBT Cross-Section Discretisation. Figure 2 shows the nodes considered in the lipped channel section. Thiscross-section discretisation leads to 17 deformation modes, which are global (1-4), distortional (5-6) and local(7-17) figure 3 shows the in-plane configurations of those more relevant for the buckling analyses.(ii) GBT Member Discretisation. The GBT equilibrium equations were solved using the beam finite elementdeveloped by Camotim et al. [2]. Each beam span was discretised into 20 finite elements for the three beams.(iii) SFE Discretisation. The SFE analyses, performed in the code ANSYS [4], were based on beam discretisationsinto fine meshes of 4-node SHELL181 elements, as illustrated in figures 4(a1), 4(b1) and 4(c1).(iv) Material Modelling. The steel material behaviour was deemed either linear elastic (bucking and post-bucklinganalyses) or linear-elastic/perfectly-plastic with von Mises yield criterion (post-buckling analyses).(v) Initial Imperfections. Critical-mode initial geometrical imperfections with amplitude equal to either10% of the wall thickness (local or distortional buckling) or L/2000 (global buckling).q200(mm)L/2L/2Natural nodeIntermediate nodeEnd nodeL 4.0, 8.0 and 10.0m172.0100(a)(b)Figure 1: Two-span continuous beam (a) cross-section dimensionsand (b) loading and first-order elastic bending moment diagram.2345678Figure 2: Lipped channelGBT discretisation.91011Figure 3: In plane shapes of the 10 most relevant lipped channel deformation modes.2.1 Buckling ResultsFigures 4(a1)-(c2) show the critical buckling mode shapes of the lipped channel B4, B8 and B10 beams theleft and right hand side figures are (i) ANSYS 3D views and (ii) GBT modal amplitude functions, respectively. Thecorresponding beam critical buckling loads, yielded by the GBT and ANSYS analyses, are (i) qcr.GBT 46.66kN/m andqcr.ANSYS 46.78kN/m (B4), (ii) qcr.GBT 10.82kN/m and qcr.ANSYS 10.71kN/m (B8), and (iii) qcr.GBT 6.06kN/mand qcr.ANSYS 5.92kN/m (B10). The analysis of these buckling results prompts the following remarks:(i) The GBT and ANSYS critical buckling loads practically coincide the maximum difference is 2.4% andconcerns the B10 beam, which buckles in a predominantly global mode. There is also very closeagreement between the buckling mode shapes – this is particularly striking if one looks at the zoomed buckled914
Cilmar Basaglia and Dinar Camotimintermediate support regions shown in figures 4(a1), (b1) and (c1).(ii) The three beam critical buckling modes combine at least two types of deformation modes: (ii1) local anddistortional (B4), and (ii2) local, distortional and global (B8 and B10).(iii) Figure 4(a2) shows that buckling involves only the B4 intermediate support region. The local modes 7-11 aredominant and the distortional modes 5-6 also play a relevant role (see the bottom flange rotation).(iv) Figure 4(b2) shows that the B8 buckling mode involves mostly the intermediate support region there is amajor contribution from the distortional modes 5-6 and also a non-negligible one from the local modes 7-8.(note that it is quite difficult to detect the latter in the ANSYS output). The global modes 3-4 (minor axis bendingand torsion) play a lesser role their contributions are only meaningful within the beam spans.(v) Figure 4(c2) shows that the global modes 3-4 are highly dominant in the B10 buckling mode they exhibit onehalf-wave per span with maximum participations at the middle. Moreover, there are small contributions of thelocal (7-8) and distortional (5-6) modes, but restricted to the close vicinity of the intermediate support.It is worth noting that, in all existing design procedures, a crucial step is the identification of the buckling modenature, which is by no means clear in the above three beams. In order to attempt to establish the “dominant nature”of these buckling modes, additional GBT analyses were carried out including only global (2-4), distortional (5-6)and local (7-17) deformation modes. Table 1 shows the relation between the “pure” global (qb.e), distortional (qb.d)and local (qb.l) buckling loads and critical buckling load (qcr), obtained with all the deformation modes the“dominant buckling mode nature”, indicated in the last column, reflects the “closeness” between the corresponding“pure” buckling load and qcr (lowest of the three ratios), and obviously agrees with results shown in figure 4.1.095L/2 2.0m60.0107-1.00.0511 81.02.03.04.06.08.07.510.0L (m)(a2)(a1)1.054 x (10)L/2 4.0m60.07 x (2)38 x (5)-1.00.02.04.0L (m)(b2)(b1)1.0568 x (5)0.0L/2 5.0m54 x (2)7 x (2)-1.00.02.55.03L (m)(c2)(c1)Figure 4: ANSYS and GBT-based buckling mode shapes of the (a1)-(a2) B4, (b1)-(b2) B8 and (c1)-(c2) B10 beams.915
Cilmar Basaglia and Dinar CamotimTable 1: Relation between qb and qcr load values.Beamqb.e / qcrqb.d / qcrqb.l / 1.391Dominant bucklingmode natureLocalDistortionalGlobal2.2 Post-Buckling ResultsThis section addresses the SFE analysis of the elastic and elastic-plastic (yield stresses fy 250, 350, 550, 850 MPa)post-buckling behaviour of the B4, B8 and B10 beams. The curves shown in figures 5(a), 6(a) and 7(a) are the postbuckling equilibrium paths q vs. V1, q vs. V2 and q vs. V3, where (i) the symbols , , and indicate the ultimateloads and (ii) V1, V2 and V3 are displacements selected to provide a better characterisation of the beam post-bucklingbehaviours they correspond to the vertical displacements of the bottom flange-lip corner of cross-sections located inthe beam left span and 23.4cm (V1) or 26.7cm (V2 and V3) away from the intermediate support (see fig. 5(b)).Figure 5(c) concerns the B4 beam with fy 250MPa and displays the deformed configurations and von Mises stressdistributions associated with (i) the full yielding of the mid-cross-section (i.e., the formation of the first beam plastichinge) and (ii) the beam collapse. As for figures 6(b) and 7(b), they provide the failure mode and von Mises stressdistributions of the B8 beam with fy 550MPa and the B10 beam with fy 850MPa, respectively.60q (kN/m)Elastic850qcr50m.4c23550350m.7c26( II )4030qu (kN/m)58.6953.2342.0237.75(I)fy 250MPa2010IntermediateV2V1 SupportV3V1 (mm)0051015End Support(a)(b)II: q 37.75kN/mI: q 33.38kN/mfy 250MPa1st plastichingeMPaMPafy 56194.537222.2192502nd plastichinge3rd 7.986195.324222.662250(c)Figure 5: B4 beam (a) equilibrium paths, (b) location of the measured displacements and (c) deformed configurationand von Mises stresses associated with the formation of the first plastic hinge and at beam collapse.916
Cilmar Basaglia and Dinar Camotim12q (kN/m)qcr85010Elastic350550MPa8qu (kN/m)11.6911.089.598.586fy 250MPa42fy 550MPaV2 367.256428.116488.976549.836(a)(b)Figure 6: B8 beam (a) equilibrium paths and (b) deformed configuration and von Mises stresses at collapse.7q (kN/m)850qcr6ElasticMPa55053504fy 250MPa3210024qu (kN/m)6.055.975.354.39 V3 (mm)68fy 51.737285.871320.005(a)(b)Figure 7: B10 beam (a) equilibrium paths and (b) deformed configuration and von Mises stresses at collapse.The analysis of the post-buckling results presented in figures 5 to 7 leads to the following conclusions:(i) The amount of post-critical strength reserve increases as (i1) the yield stress increases (obviously) and (i2) one“travels” from the B10 beams to the B4 ones, due to the growing presence of local buckling. The higher postcritical strength reserve occurs for the B4 beam with fy 850MPa ultimate-to-critical load ratio equal to 1.25.(ii) The bending moment redistribution is very clear in the beams with low yield stresses, whose collapse is lessaffected by the geometrically non-linear (buckling) effects. Figure 5(c) shows the deformed configurations ofthe B4 beam with fy 250MPa associated with (ii1) the full yielding of the intermediate support cross-section,occurring for q 33.38kN/m (point I), and (ii2) the beam collapse, occurring for q 37.75kN/m (point II) andcorresponding to the nearly simultaneously yielding of cross-section near the mid-spans.(iii) The failure mechanisms of the beams with high yield stresses are very similar to corresponding critical bucklingmodes, thus implying that the collapse stems mainly from geometrically non-linear effects. Moreover, it isworth noting that the failures of all the beams with fy 850MPa occur practically in the elastic range.(iv) Increasing the yield stresses from 250MPa to 850MPa leads to ultimate load increases of 55.5% (B4 beams),36.2% (B8 beams) and 37.8% (B10 beams).3 DSM DESIGN PROCEDUREThe DSM adopts “Winter-type” design curves, calibrated against experimental and numerical results concerningthe ultimate strength of isolated (single-span) members acted by uniform internal force/moment. In beams, thenominal bending strengths against local (Mnl), distortional (Mnd) and global (Mne) failures are given by the expressions917
Cilmar Basaglia and Dinar CamotimM nl M y§§M M nl 1 0.15 crl My ( λl M y / M crl 0.776 )M nd M y( λe M y / M cre 0.60 )0.4 · § M crl My¹ 0.4· My ¹(1)( λl 0.776 )M nd§§M 1 0.22 crd My ( λd M y / M crd 0.673 )M ne M y· ¹0.5 ·· ¹ § M crd My¹ 0.5· My ¹(2)( λd 0.673 )M ne 10 M y10 § 1 9 36 M cre(0.60 λe 1.336)· M y¹M ne M cre, (3)(λe 1.336)where (i) λl, λd and λe are local, distortional and global slenderness values, (ii) Mcrl, Mcrd and Mcre are the elasticglobal, local and distortional critical buckling moments and (iii) M y W y f y is the cross-section first yield moment Wy is the elastic modulus. In beams subjected to non-uniform bending, it is convenient to replace the various “Myvalues” appearing in (1)-(3) by “first yield load parameter values” in this case, the obvious choice is qy 32 My/L2.It is worth noting that the application of expressions (1)-(3) corresponds to neglecting (i) the cross-section elasticplastic strength reserve, in statically determinate or indeterminate beams, and (ii) the bending moment redistribution,in statically indeterminate beams this means that overly conservative predictions are to be expected in staticallyindeterminate beams, particularly in the lower slenderness range.Figures 8 to 10 show comparisons between the ultimate load predictions yielded by the current DSM designcurves and the collapse loads obtained through SFE analyses involving B4, B8 and B10 beams with 15 different yieldstresses, associated with yield-to-critical load ratios qy /qcr ranging from 0.06 to 3.74. The numerical ultimate loads,normalised w.r.t. qy, are represented by the symbols , and , respectively for beam local (B4), distortional(B8) and global (B10) failures. Since the beams exhibit buckling and failure modes that are not “pure”, the DSMcurve choice was made on the basis of their “dominant buckling mode nature”, given in table 1 however, the λl, λdand λe are calculated with the actual beam critical buckling load qcr, which is neither “purely” local, distortional orglobal. The observation of these comparisons prompts the following remarks:(i) The DSM predictions are (i1) excessively safe in the low slenderness range, (i2) slightly safe in the intermediateslenderness range and (i3) unsafe (local and distortional) or accurate (global) in the high slenderness range.(ii) None of the DSM curves can predict efficiently (safely and economically) the two-span beam collapse loads,which is due to a combination of (ii1) neglecting both the cross-section elastic-plastic strength reserve and(mostly) the bending moment redistribution (low slenderness range) and (ii2) the “mixed” nature of the failuremechanisms (high slenderness range).(iii) In the high slenderness range, the elastic critical buckling load curves (dashed lines) are either slightly below (B4)or passes right through (B8 and B10) the beam collapse load ratios.(iv) Since the beam collapse loads already incorporate the local, distortional and global buckling effects, it seems tomake little sense to neglect the cross-section elastic-plastic strength reserve and beam moment redistribution. Therecent work by Shifferaw and Schafer [5] confirms this assertion it reports experimental and numericalevidence, involving simply supported isolated beams (no moment redistribution), of the (logical) presence of anon-negligible inelastic strength in the low slenderness range.(v) The most rational approach to account for the beam inelastic strength reserve (including moment redistribution)is to replace qy (first yield loads) by qpl (geometrically linear plastic collapse loads of the whole beam) in (1)-(3).Figures 8 to 10 also show comparisons between the ultimate load predictions yielded by these modified DSMdesign curves and the previous SFE collapse loads, now normalised w.r.t. qpl and represented by the symbols, and . The observation of these new comparisons leads to the following comments:918
Cilmar Basaglia and Dinar Camotim(v.1) In the low slenderness range, the modified DSM predictions are either very accurate (local and distortional)or barely unsafe (global), which confirms the presence and relevance of the beam inelastic strength reserve.(v.2) In the intermediate slenderness range, the modified DSM predictions are either accurate-to-unsafe (localand distortional) or clearly unsafe (global).(v.3) In the high slenderness range, the modified DSM predictions practically coincide with the previous ones.1.6DSM curveElastic buckling(qu /qy)(qu /qpl)1.41.2§ qu qy · 1.0 ¹§ qu q pl · 0.6 ¹ 0.40.80.20.00.00.40.81.21.62.02.4(qy /qcrl)0.5 or (qpl /qcrl)0.5Figure 8: Comparison between SFE B4 beam collapse loads and DSM local design curve predictions.1.6DSM curveElastic buckling(qu /qy)(qu /qpl)1.41.2§ qu qy · 1.0 ¹§ qu q pl · 0.6 ¹ 0.40.80.20.00.00.40.81.21.62.02.4(qy /qcrd)0.5 or (qpl /qcrd)0.5Figure 9: Comparison between SFE B8 beam collapse loads and DSM distortional design curve predictions.1.6DSM curveElastic buckling(qu /qy)(qu /qpl)1.41.2§ qu qy · 1.0 ¹§ qu q pl · 0.6 ¹ 0.40.80.20.00.00.40.81.21.62.02.4(qy /qcre)0.5 or (qpl /qcre)0.5Figure 10: Comparison between SFE B10 beam collapse loads and DSM global design curve predictions.Although much more research work is obviously needed before it is possible to have a firm opinion on the DSMdesign of multi-span cold-formed steel beam, it seems possible to make some preliminary comments on the basis of919
Cilmar Basaglia and Dinar Camotimthe limited amount of results (both in quantity and in scope) presented in this work:(i) Since there are no “pure” buckling and failure modes, the DSM curve choice should be based on the concept of“dominant buckling/failure mode nature”.(ii) The first yield load (moment) should be replaced by the first-order plastic collapse load (moment), thusaccounting for the cross-section elastic-plastic strength reserve and beam moment redistribution. Failing to dothis will inevitably lead to overly conservative prediction in the low slenderness range.(iii) Apparently, the most rational approach is to develop and calibrate design curves that are based on (iii1) theplastic collapse load, for stocky beams, and on (iii2) the elastic buckling load, for slender beams. Nothing can yetbe said about the intermediate slenderness range (or about the slenderness limits separating the three ranges) nevertheless, the current DSM design curves provide the quite satisfactory ultimate load estimates in this range.4 CONCLUSIONThis work reported the results of an ongoing numerical investigation on the buckling, post-buckling, collapseand design of two-span cold-formed steel lipped channel beams subjected to uniformly distributed loads. Theseresults consisted of (i) critical buckling loads and mode shapes, determined through GBT and ANSYS analyses,(ii) post-buckling equilibrium paths (up to collapse), deformed configurations and von Mises stress distributions,obtained by means of ANSYS elastic and elastic-plastic shell finite element analyses, and (iii) ultimate loadpredictions, yielded by the current DSM design curves. The following aspects deserve to be mentioned:(i) The beam buckling and failure modes combine two or three types of deformation modes, which precludes astraightforward classification. Thus, one must resort to the “dominant buckling/failure mode nature” concept.(ii) The beam post-buckling behaviour and inelastic strength reserve (ii1) depend on the buckling/failure modecharacteristics and yield-to-critical stress ratio, and (ii2) may be heavily affected by moment redistribution,provided that the yield stress is low enough on the other hand, failure may occur in the elastic range in beamswith high yield stresses.(iii) Due to the “mixed nature” of the beam failure modes, the choice of the appropriate DSM design curve (amongstthe currently available ones) must also be based on the “dominant buckling/failure mode nature” concept.(iv) The direct application of the current DSM design curves leads to either over-conservative (low slenderness) orclearly unsafe (high slenderness) beam ultimate load predictions.(v) The numerical ultimate loads obtained clearly indicate that (v1) the beams with low slenderness exhibit a fairamount of inelastic strength reserve, stemming mostly from the moment redistribution, and (v2) ultimate loads ofbeams with high slenderness are fairly well approximated by their critical buckling loads, particularly if globalbuckling is involved. Although further studies are required to confirm these preliminary findings, it seems thatthe current DSM design curves will only be efficient if modified to take into account these two aspects.ACKNOWLEDGEMENTSThe first author gratefully acknowledges the financial support provided by “Fundação para a Ciência eTecnologia” (FCT Portugal), through the post-doctoral scholarship nº er B.W., “Review: the direct strength method of cold-formed steel member design”, Journal ofConstructional Steel Research, 64(7-8), 766-778, 2008.Camotim D., Silvestre N., Basaglia C. and Bebiano R., “GBT-based buckling analysis of thin-walled memberswith non-standard support conditions”, Thin-Walled Structures, 46(7-9), 800-815, 2008.Yu C. and Schafer B.W., “Simulation of cold-formed steel beams in local and distortional buckling withapplications to the direct strength method”, Journal of Constructional Steel Research, 63(5), 581-590, 2007.Swanson Analysis Systems Inc., ANSYS Reference Manual (version 8.1), 2004.Shifferaw Y. and Schafer B.W., “Inelastic bending capacity in cold-formed steel members”, Proceedings ofStructural Stability Research Council Annual Stability Conference (New Orleans, 18-21/4), 279-299, 2007.920
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